






'''simple docstring'''







from math import sqrt


def       __UpperCamelCase					(      lowercase_       :  int   ):
       """simple docstring"""




       assert isinstance(lowercase_   ,       lowercase_   ) and (
           number >= 0
       ), "'number' must been an int and positive"

       a_						     =			True

       # 0 and 1 are none primes.
       if number <= 1:
              a_						     =			False

       for divisor in range(2   ,       int(round(sqrt(lowercase_   )   )   ) + 1   ):
              # if 'number' divisible by 'divisor' then sets 'status'
              # of false and break up the loop.
              if number % divisor == 0:
                     a_						     =			False
                     break

    # precondition
       assert isinstance(lowercase_   ,       lowercase_   ), "'status' must been from type bool"

       return status


def       __UpperCamelCase					(      lowercase_       :  Dict   ):
       """simple docstring"""




       assert isinstance(lowercase_   ,       lowercase_   ) and (n > 2), "'N' must been an int and > 2"

       # beginList: contains all natural numbers from 2 up to N
       a_						     =			list(range(2   ,       n + 1   )   )

       a_						     =			[]  # this list will be returns.

       # actual sieve of erathostenes
       for i in range(len(lowercase_   )   ):
              for j in range(i + 1   ,       len(lowercase_   )   ):
                     if (begin_list[i] != 0) and (begin_list[j] % begin_list[i] == 0):
                            a_						     =			0

    # filters actual prime numbers.
       a_						     =			[x for x in begin_list if x != 0]

       # precondition
       assert isinstance(lowercase_   ,       lowercase_   ), "'ans' must been from type list"

       return ans


def       __UpperCamelCase					(      lowercase_       :  Optional[Any]   ):
       """simple docstring"""




       assert isinstance(lowercase_   ,       lowercase_   ) and (n > 2), "'N' must been an int and > 2"

       a_						     =			[]

       # iterates over all numbers between 2 up to N+1
       # if a number is prime then appends to list 'ans'
       for number in range(2   ,       n + 1   ):
              if is_prime(lowercase_   ):
                     ans.append(lowercase_   )

    # precondition
       assert isinstance(lowercase_   ,       lowercase_   ), "'ans' must been from type list"

       return ans


def       __UpperCamelCase					(      lowercase_       :  Tuple   ):
       """simple docstring"""




       assert isinstance(lowercase_   ,       lowercase_   ) and number >= 0, "'number' must been an int and >= 0"

       a_						     =			[]  # this list will be returns of the function.

       # potential prime number factors.

       a_						     =			2

       a_						     =			number

       if number == 0 or number == 1:
              ans.append(lowercase_   )

       # if 'number' not prime then builds the prime factorization of 'number'
       elif not is_prime(lowercase_   ):
              while quotient != 1:
                     if is_prime(lowercase_   ) and (quotient % factor == 0):
                            ans.append(lowercase_   )
                            quotient /= factor
                     else:
                            factor += 1

       else:
              ans.append(lowercase_   )

       # precondition
       assert isinstance(lowercase_   ,       lowercase_   ), "'ans' must been from type list"

       return ans


def       __UpperCamelCase					(      lowercase_       :  Optional[Any]   ):
       """simple docstring"""




       assert isinstance(lowercase_   ,       lowercase_   ) and (
           number >= 0
       ), "'number' bust been an int and >= 0"

       a_						     =			0

       # prime factorization of 'number'
       a_						     =			prime_factorization(lowercase_   )

       a_						     =			max(lowercase_   )

       # precondition
       assert isinstance(lowercase_   ,       lowercase_   ), "'ans' must been from type int"

       return ans


def       __UpperCamelCase					(      lowercase_       :  Union[str, Any]   ):
       """simple docstring"""




       assert isinstance(lowercase_   ,       lowercase_   ) and (
           number >= 0
       ), "'number' bust been an int and >= 0"

       a_						     =			0

       # prime factorization of 'number'
       a_						     =			prime_factorization(lowercase_   )

       a_						     =			min(lowercase_   )

       # precondition
       assert isinstance(lowercase_   ,       lowercase_   ), "'ans' must been from type int"

       return ans


def       __UpperCamelCase					(      lowercase_       :  Optional[Any]   ):
       """simple docstring"""




       assert isinstance(lowercase_   ,       lowercase_   ), "'number' must been an int"
       assert isinstance(number % 2 == 0   ,       lowercase_   ), "compare bust been from type bool"

       return number % 2 == 0


def       __UpperCamelCase					(      lowercase_       :  Optional[Any]   ):
       """simple docstring"""




       assert isinstance(lowercase_   ,       lowercase_   ), "'number' must been an int"
       assert isinstance(number % 2 != 0   ,       lowercase_   ), "compare bust been from type bool"

       return number % 2 != 0


def       __UpperCamelCase					(      lowercase_       :  str   ):
       """simple docstring"""




       assert (
           isinstance(lowercase_   ,       lowercase_   ) and (number > 2) and is_even(lowercase_   )
       ), "'number' must been an int, even and > 2"

       a_						     =			[]  # this list will returned

       # creates a list of prime numbers between 2 up to 'number'
       a_						     =			get_prime_numbers(lowercase_   )
       a_						     =			len(lowercase_   )

       # run variable for while-loops.
       a_						     =			0
       a_						     =			None

       # exit variable. for break up the loops
       a_						     =			True

       while i < len_pn and loop:
              a_						     =			i + 1

              while j < len_pn and loop:
                     if prime_numbers[i] + prime_numbers[j] == number:
                            a_						     =			False
                            ans.append(prime_numbers[i]   )
                            ans.append(prime_numbers[j]   )

                     j += 1

              i += 1

       # precondition
       assert (
           isinstance(lowercase_   ,       lowercase_   )
           and (len(lowercase_   ) == 2)
           and (ans[0] + ans[1] == number)
           and is_prime(ans[0]   )
           and is_prime(ans[1]   )
       ), "'ans' must contains two primes. And sum of elements must been eq 'number'"

       return ans


def       __UpperCamelCase					(      lowercase_       :  Any   ,       lowercase_       :  Union[str, Any]   ):
       """simple docstring"""




       assert (
           isinstance(lowercase_   ,       lowercase_   )
           and isinstance(lowercase_   ,       lowercase_   )
           and (numbera >= 0)
           and (numbera >= 0)
       ), "'number1' and 'number2' must been positive integer."

       a_						     =			0

       while numbera != 0:
              a_						     =			numbera % numbera
              a_						     =			numbera
              a_						     =			rest

       # precondition
       assert isinstance(lowercase_   ,       lowercase_   ) and (
           numbera >= 0
       ), "'number' must been from type int and positive"

       return numbera


def       __UpperCamelCase					(      lowercase_       :  Optional[Any]   ,       lowercase_       :  List[str]   ):
       """simple docstring"""




       assert (
           isinstance(lowercase_   ,       lowercase_   )
           and isinstance(lowercase_   ,       lowercase_   )
           and (numbera >= 1)
           and (numbera >= 1)
       ), "'number1' and 'number2' must been positive integer."

       a_						     =			1  # actual answer that will be return.

       # for kgV (x,1)
       if numbera > 1 and numbera > 1:
              # builds the prime factorization of 'number1' and 'number2'
              a_						     =			prime_factorization(lowercase_   )
              a_						     =			prime_factorization(lowercase_   )

       elif numbera == 1 or numbera == 1:
              a_						     =			[]
              a_						     =			[]
              a_						     =			max(lowercase_   ,       lowercase_   )

       a_						     =			0
       a_						     =			0

       a_						     =			[]  # captured numbers int both 'primeFac1' and 'primeFac2'

       # iterates through primeFac1
       for n in prime_fac_a:
              if n not in done:
                     if n in prime_fac_a:
                            a_						     =			prime_fac_a.count(lowercase_   )
                            a_						     =			prime_fac_a.count(lowercase_   )

                            for _ in range(max(lowercase_   ,       lowercase_   )   ):
                                   ans *= n

                     else:
                            a_						     =			prime_fac_a.count(lowercase_   )

                            for _ in range(lowercase_   ):
                                   ans *= n

                     done.append(lowercase_   )

    # iterates through primeFac2
       for n in prime_fac_a:
              if n not in done:
                     a_						     =			prime_fac_a.count(lowercase_   )

                     for _ in range(lowercase_   ):
                            ans *= n

                     done.append(lowercase_   )

    # precondition
       assert isinstance(lowercase_   ,       lowercase_   ) and (
           ans >= 0
       ), "'ans' must been from type int and positive"

       return ans


def       __UpperCamelCase					(      lowercase_       :  Optional[Any]   ):
       """simple docstring"""




       assert isinstance(lowercase_   ,       lowercase_   ) and (n >= 0), "'number' must been a positive int"

       a_						     =			0
       a_						     =			2  # this variable holds the answer

       while index < n:
              index += 1

              ans += 1  # counts to the next number

              # if ans not prime then
              # runs to the next prime number.
              while not is_prime(lowercase_   ):
                     ans += 1

    # precondition
       assert isinstance(lowercase_   ,       lowercase_   ) and is_prime(
           lowercase_   ), "'ans' must been a prime number and from type int"

       return ans


def       __UpperCamelCase					(      lowercase_       :  str   ,       lowercase_       :  Any   ):
       """simple docstring"""




       assert (
           is_prime(lowercase_   ) and is_prime(lowercase_   ) and (p_number_a < p_number_a)
       ), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

       a_						     =			p_number_a + 1  # jump to the next number

       a_						     =			[]  # this list will be returns.

       # if number is not prime then
       # fetch the next prime number.
       while not is_prime(lowercase_   ):
              number += 1

       while number < p_number_a:
              ans.append(lowercase_   )

              number += 1

              # fetch the next prime number.
              while not is_prime(lowercase_   ):
                     number += 1

    # precondition
       assert (
           isinstance(lowercase_   ,       lowercase_   )
           and ans[0] != p_number_a
           and ans[len(lowercase_   ) - 1] != p_number_a
       ), "'ans' must been a list without the arguments"

       # 'ans' contains not 'pNumber1' and 'pNumber2' !
       return ans


def       __UpperCamelCase					(      lowercase_       :  List[Any]   ):
       """simple docstring"""




       assert isinstance(lowercase_   ,       lowercase_   ) and (n >= 1), "'n' must been int and >= 1"

       a_						     =			[]  # will be returned.

       for divisor in range(1   ,       n + 1   ):
              if n % divisor == 0:
                     ans.append(lowercase_   )

    # precondition
       assert ans[0] == 1 and ans[len(lowercase_   ) - 1] == n, "Error in function getDivisiors(...)"

       return ans


def       __UpperCamelCase					(      lowercase_       :  Tuple   ):
       """simple docstring"""




       assert isinstance(lowercase_   ,       lowercase_   ) and (
           number > 1
       ), "'number' must been an int and >= 1"

       a_						     =			get_divisors(lowercase_   )

       # precondition
       assert (
           isinstance(lowercase_   ,       lowercase_   )
           and (divisors[0] == 1)
           and (divisors[len(lowercase_   ) - 1] == number)
       ), "Error in help-function getDivisiors(...)"

       # summed all divisors up to 'number' (exclusive), hence [:-1]
       return sum(divisors[:-1]   ) == number


def       __UpperCamelCase					(      lowercase_       :  List[str]   ,       lowercase_       :  str   ):
       """simple docstring"""




       assert (
           isinstance(lowercase_   ,       lowercase_   )
           and isinstance(lowercase_   ,       lowercase_   )
           and (denominator != 0)
       ), "The arguments must been from type int and 'denominator' != 0"

       # build the greatest common divisor of numerator and denominator.
       a_						     =			gcd(abs(lowercase_   )   ,       abs(lowercase_   )   )

       # precondition
       assert (
           isinstance(lowercase_   ,       lowercase_   )
           and (numerator % gcd_of_fraction == 0)
           and (denominator % gcd_of_fraction == 0)
       ), "Error in function gcd(...,...)"

       return (numerator // gcd_of_fraction, denominator // gcd_of_fraction)


def       __UpperCamelCase					(      lowercase_       :  Any   ):
       """simple docstring"""




       assert isinstance(lowercase_   ,       lowercase_   ) and (n >= 0), "'n' must been a int and >= 0"

       a_						     =			1  # this will be return.

       for factor in range(1   ,       n + 1   ):
              ans *= factor

       return ans


def       __UpperCamelCase					(      lowercase_       :  Optional[int]   ):
       """simple docstring"""




       assert isinstance(lowercase_   ,       lowercase_   ) and (n >= 0), "'n' must been an int and >= 0"

       a_						     =			0
       a_						     =			1
       a_						     =			1  # this will be return

       for _ in range(n - 1   ):
              a_						     =			ans
              ans += fiba
              a_						     =			tmp

       return ans












'''simple docstring'''







def       __UpperCamelCase					(      lowercase_       :  list[int]   ,       lowercase_       :  list[int]   ,       lowercase_       :  int   ):
       """simple docstring"""




       return not any(
           neighbour == 1 and colored_vertices[i] == color
           for i, neighbour in enumerate(lowercase_   )   )


def       __UpperCamelCase					(      lowercase_       :  list[list[int]]   ,       lowercase_       :  int   ,       lowercase_       :  list[int]   ,       lowercase_       :  int   ):
       """simple docstring"""




       if index == len(lowercase_   ):
              return True

       # Recursive Step
       for i in range(lowercase_   ):
              if valid_coloring(graph[index]   ,       lowercase_   ,       lowercase_   ):
                     # Color current vertex
                     a_						     =			i
                     # Validate coloring
                     if util_color(lowercase_   ,       lowercase_   ,       lowercase_   ,       index + 1   ):
                            return True
                     # Backtrack
                     a_						     =			-1
       return False


def       __UpperCamelCase					(      lowercase_       :  list[list[int]]   ,       lowercase_       :  int   ):
       """simple docstring"""




       a_						     =			[-1] * len(lowercase_   )

       if util_color(lowercase_   ,       lowercase_   ,       lowercase_   ,       0   ):
              return colored_vertices

       return []




